Document Type
Article
Publication Date
10-1-2018
Abstract
We obtain necessary and sufficient conditions with sharp constants on the distribution (Formula presented.) for the existence of a globally finite energy solution to the quasi-linear equation with a gradient source term of natural growth of the form (Formula presented.) in a bounded open set (Formula presented.). Here (Formula presented.), (Formula presented.), is the standard (Formula presented.) -Laplacian operator defined by (Formula presented.). The class of solutions that we are interested in consists of functions (Formula presented.) such that (Formula presented.) for some (Formula presented.) and the inequality (Formula presented.) holds for all (Formula presented.) with some constant (Formula presented.). This is a natural class of solutions at least when the distribution (Formula presented.) is nonnegative. The study of (Formula presented.) is applied to show the existence of globally finite energy solutions to the quasi-linear equation of Schrödinger type (Formula presented.), (Formula presented.) in (Formula presented.), and (Formula presented.) on (Formula presented.), via the exponential transformation (Formula presented.).
Publication Source (Journal or Book title)
Journal of the London Mathematical Society
First Page
461
Last Page
482
Recommended Citation
Adimurthi, K., & Phuc, N. (2018). Nonlinear equations with gradient natural growth and distributional data, with applications to a Schrödinger type equation. Journal of the London Mathematical Society, 98 (2), 461-482. https://doi.org/10.1112/jlms.12143